3.1791 \(\int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx\)

Optimal. Leaf size=92 \[ -\frac {81}{64} (1-2 x)^{15/2}+\frac {81}{4} (1-2 x)^{13/2}-\frac {97335}{704} (1-2 x)^{11/2}+\frac {4165}{8} (1-2 x)^{9/2}-\frac {74235}{64} (1-2 x)^{7/2}+\frac {12005}{8} (1-2 x)^{5/2}-\frac {184877}{192} (1-2 x)^{3/2} \]

[Out]

-184877/192*(1-2*x)^(3/2)+12005/8*(1-2*x)^(5/2)-74235/64*(1-2*x)^(7/2)+4165/8*(1-2*x)^(9/2)-97335/704*(1-2*x)^
(11/2)+81/4*(1-2*x)^(13/2)-81/64*(1-2*x)^(15/2)

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Rubi [A]  time = 0.02, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ -\frac {81}{64} (1-2 x)^{15/2}+\frac {81}{4} (1-2 x)^{13/2}-\frac {97335}{704} (1-2 x)^{11/2}+\frac {4165}{8} (1-2 x)^{9/2}-\frac {74235}{64} (1-2 x)^{7/2}+\frac {12005}{8} (1-2 x)^{5/2}-\frac {184877}{192} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^5*(3 + 5*x),x]

[Out]

(-184877*(1 - 2*x)^(3/2))/192 + (12005*(1 - 2*x)^(5/2))/8 - (74235*(1 - 2*x)^(7/2))/64 + (4165*(1 - 2*x)^(9/2)
)/8 - (97335*(1 - 2*x)^(11/2))/704 + (81*(1 - 2*x)^(13/2))/4 - (81*(1 - 2*x)^(15/2))/64

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \sqrt {1-2 x} (2+3 x)^5 (3+5 x) \, dx &=\int \left (\frac {184877}{64} \sqrt {1-2 x}-\frac {60025}{8} (1-2 x)^{3/2}+\frac {519645}{64} (1-2 x)^{5/2}-\frac {37485}{8} (1-2 x)^{7/2}+\frac {97335}{64} (1-2 x)^{9/2}-\frac {1053}{4} (1-2 x)^{11/2}+\frac {1215}{64} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {184877}{192} (1-2 x)^{3/2}+\frac {12005}{8} (1-2 x)^{5/2}-\frac {74235}{64} (1-2 x)^{7/2}+\frac {4165}{8} (1-2 x)^{9/2}-\frac {97335}{704} (1-2 x)^{11/2}+\frac {81}{4} (1-2 x)^{13/2}-\frac {81}{64} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.47 \[ -\frac {1}{33} (1-2 x)^{3/2} \left (2673 x^6+13365 x^5+29565 x^4+38220 x^3+32220 x^2+18696 x+7288\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^5*(3 + 5*x),x]

[Out]

-1/33*((1 - 2*x)^(3/2)*(7288 + 18696*x + 32220*x^2 + 38220*x^3 + 29565*x^4 + 13365*x^5 + 2673*x^6))

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fricas [A]  time = 0.59, size = 44, normalized size = 0.48 \[ \frac {1}{33} \, {\left (5346 \, x^{7} + 24057 \, x^{6} + 45765 \, x^{5} + 46875 \, x^{4} + 26220 \, x^{3} + 5172 \, x^{2} - 4120 \, x - 7288\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/33*(5346*x^7 + 24057*x^6 + 45765*x^5 + 46875*x^4 + 26220*x^3 + 5172*x^2 - 4120*x - 7288)*sqrt(-2*x + 1)

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giac [A]  time = 1.20, size = 106, normalized size = 1.15 \[ \frac {81}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {81}{4} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {97335}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {4165}{8} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {74235}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {12005}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {184877}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

81/64*(2*x - 1)^7*sqrt(-2*x + 1) + 81/4*(2*x - 1)^6*sqrt(-2*x + 1) + 97335/704*(2*x - 1)^5*sqrt(-2*x + 1) + 41
65/8*(2*x - 1)^4*sqrt(-2*x + 1) + 74235/64*(2*x - 1)^3*sqrt(-2*x + 1) + 12005/8*(2*x - 1)^2*sqrt(-2*x + 1) - 1
84877/192*(-2*x + 1)^(3/2)

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maple [A]  time = 0.00, size = 40, normalized size = 0.43 \[ -\frac {\left (2673 x^{6}+13365 x^{5}+29565 x^{4}+38220 x^{3}+32220 x^{2}+18696 x +7288\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{33} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^5*(5*x+3)*(-2*x+1)^(1/2),x)

[Out]

-1/33*(2673*x^6+13365*x^5+29565*x^4+38220*x^3+32220*x^2+18696*x+7288)*(-2*x+1)^(3/2)

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maxima [A]  time = 0.49, size = 64, normalized size = 0.70 \[ -\frac {81}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {81}{4} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {97335}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {4165}{8} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {74235}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {12005}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {184877}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^5*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-81/64*(-2*x + 1)^(15/2) + 81/4*(-2*x + 1)^(13/2) - 97335/704*(-2*x + 1)^(11/2) + 4165/8*(-2*x + 1)^(9/2) - 74
235/64*(-2*x + 1)^(7/2) + 12005/8*(-2*x + 1)^(5/2) - 184877/192*(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 64, normalized size = 0.70 \[ \frac {12005\,{\left (1-2\,x\right )}^{5/2}}{8}-\frac {184877\,{\left (1-2\,x\right )}^{3/2}}{192}-\frac {74235\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {4165\,{\left (1-2\,x\right )}^{9/2}}{8}-\frac {97335\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {81\,{\left (1-2\,x\right )}^{13/2}}{4}-\frac {81\,{\left (1-2\,x\right )}^{15/2}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^5*(5*x + 3),x)

[Out]

(12005*(1 - 2*x)^(5/2))/8 - (184877*(1 - 2*x)^(3/2))/192 - (74235*(1 - 2*x)^(7/2))/64 + (4165*(1 - 2*x)^(9/2))
/8 - (97335*(1 - 2*x)^(11/2))/704 + (81*(1 - 2*x)^(13/2))/4 - (81*(1 - 2*x)^(15/2))/64

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sympy [A]  time = 3.22, size = 82, normalized size = 0.89 \[ - \frac {81 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {81 \left (1 - 2 x\right )^{\frac {13}{2}}}{4} - \frac {97335 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {4165 \left (1 - 2 x\right )^{\frac {9}{2}}}{8} - \frac {74235 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {12005 \left (1 - 2 x\right )^{\frac {5}{2}}}{8} - \frac {184877 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**5*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

-81*(1 - 2*x)**(15/2)/64 + 81*(1 - 2*x)**(13/2)/4 - 97335*(1 - 2*x)**(11/2)/704 + 4165*(1 - 2*x)**(9/2)/8 - 74
235*(1 - 2*x)**(7/2)/64 + 12005*(1 - 2*x)**(5/2)/8 - 184877*(1 - 2*x)**(3/2)/192

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